WIAS Preprint No. 764, (2002)

On multichannel signal detection



Authors

  • Ingster, Yuri I.
  • Lepskii, Oleg V.

2010 Mathematics Subject Classification

  • 62G10 62G20

Keywords

  • Multichannel signal detection, minimax hypothesis testing, adaptive hypothesis testing, distinguishability conditions.

Abstract

We consider $n$-channel signal detection system. Each $i$th channel could contain (or not contain) a signal. We suppose a signal is a function $f_i(t), tin (0,1)$ observing in the white Gaussian noise of level $e>0$. Let $k$ be a number of channels which contain the signals. This number could be known or unknown. The functions $f_i$ could be known or unknown as well. If shapes of functions $f_i$ are unknown, then we consider nonparametric case. We suppose that functions $f_i$ belongs to the Sobolev ball $S^sigma$ where the smoothness parameter $sigma>0$ could be known or unknown as well. The cases, when $k$ or $sigma$ are unknown, lead to the ädaptive" problems. We are interested in the following problems:

  1. How large the signals $f_i$ should be in order to detect these signals with vanishing errors, as the number of channels $n$ tends to infinity?
  2. What are the structures of test procedures which provide the detection of signals with the vanishing errors, if it is possible?
We show that there are two main types of results in the problems which, roughly, correspond to the cases either $k$ is "large" (this means $kgg n^1/2$ in the problem) or $k$ is ßmall" (this means $kll n^1/2$).

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